8269edo - Revision history

FloraC: - unuseful data. Highlight important stuff

2026-05-18T17:44:47Z

- unuseful data. Highlight important stuff

← Older revision		Revision as of 17:44, 18 May 2026
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	== Theory ==		== Theory ==
	8269edo ~~is both a [[The Riemann zeta function and tuning #Zeta edo lists\|zeta peak and zeta integral edo]], which has to do with the fact that it~~ is a very strong [[19-limit\|19-]] and [[23-limit]] system~~. It has~~ a lower 19-limit and a lower 23-limit [[~~Tenney-Euclidean~~ temperament measures #TE simple badness\|relative error]] than any smaller ~~division,~~ a ~~lower 19-limit~~ [[~~Tenney-Euclidean temperament measures~~ #~~TE simple badness~~\|~~TE logflat badness]] than any smaller division,~~ and ~~a lower 23-limit logflat badness than any excepting [[311edo\|311]], [[581edo\|581]], [[1578edo\|1578]] and [[2460edo\|2460~~]]. While [[8539edo\|8539]] has received most of the attention in this size range, 8269 is actually a bit better in the 23-limit and nearly as good in the 19-limit. They are rather like twins, including the fact both are primes. The sum of 8269 and 8539 is [[16808edo\|16808]], which does well in not just the 23-limit, but also the 31-limit. The difference between 8269 and 8539 is [[270edo\|270]], which does well in the 13-limit, and also beyond. A step of 8269edo has also been similarly proposed as an [[interval size measure]], the '''major tina'''.		8269edo is a very strong [[19-limit\|19-]] and [[23-limit]] system, with a lower 19-limit and a lower 23-limit [[Tenney–Euclidean temperament measures #TE simple badness\|relative error]] than any smaller edo. It is both a [[the Riemann zeta function and tuning #Zeta edo lists\|zeta peak and zeta integral edo]].

			While [[8539edo\|8539]] has received most of the attention in this size range, 8269 is actually a bit better in the 23-limit and nearly as good in the 19-limit. They are rather like twins, including the fact both are primes. The sum of 8269 and 8539 is [[16808edo\|16808]], which does well in not just the 23-limit, but also the 31-limit. The difference between 8269 and 8539 is [[270edo\|270]], which does well in the 13-limit, and also beyond. A step of 8269edo has also been similarly proposed as an [[interval size measure]], the '''major tina'''.

	Some of the simpler commas it [[tempering out\|tempers out]] include [[123201/123200]] in the 13-limit; [[194481/194480]], [[336141/336140]] in the 17-limit; 23409/23408, 28900/28899, 43681/43680, 89376/89375 in the 19-limit; and 21505/21504 among others in the 23-limit.		Some of the simpler commas it [[tempering out\|tempers out]] include [[123201/123200]] in the 13-limit; [[194481/194480]], [[336141/336140]] in the 17-limit; 23409/23408, 28900/28899, 43681/43680, 89376/89375 in the 19-limit; and 21505/21504 among others in the 23-limit.

FloraC: Fix math error

2026-05-18T17:40:19Z

Fix math error

← Older revision		Revision as of 17:40, 18 May 2026
Line 5:		Line 5:
	8269edo is both a [[The Riemann zeta function and tuning #Zeta edo lists\|zeta peak and zeta integral edo]], which has to do with the fact that it is a very strong [[19-limit\|19-]] and [[23-limit]] system. It has a lower 19-limit and a lower 23-limit [[Tenney-Euclidean temperament measures #TE simple badness\|relative error]] than any smaller division, a lower 19-limit [[Tenney-Euclidean temperament measures #TE simple badness\|TE logflat badness]] than any smaller division, and a lower 23-limit logflat badness than any excepting [[311edo\|311]], [[581edo\|581]], [[1578edo\|1578]] and [[2460edo\|2460]]. While [[8539edo\|8539]] has received most of the attention in this size range, 8269 is actually a bit better in the 23-limit and nearly as good in the 19-limit. They are rather like twins, including the fact both are primes. The sum of 8269 and 8539 is [[16808edo\|16808]], which does well in not just the 23-limit, but also the 31-limit. The difference between 8269 and 8539 is [[270edo\|270]], which does well in the 13-limit, and also beyond. A step of 8269edo has also been similarly proposed as an [[interval size measure]], the '''major tina'''.		8269edo is both a [[The Riemann zeta function and tuning #Zeta edo lists\|zeta peak and zeta integral edo]], which has to do with the fact that it is a very strong [[19-limit\|19-]] and [[23-limit]] system. It has a lower 19-limit and a lower 23-limit [[Tenney-Euclidean temperament measures #TE simple badness\|relative error]] than any smaller division, a lower 19-limit [[Tenney-Euclidean temperament measures #TE simple badness\|TE logflat badness]] than any smaller division, and a lower 23-limit logflat badness than any excepting [[311edo\|311]], [[581edo\|581]], [[1578edo\|1578]] and [[2460edo\|2460]]. While [[8539edo\|8539]] has received most of the attention in this size range, 8269 is actually a bit better in the 23-limit and nearly as good in the 19-limit. They are rather like twins, including the fact both are primes. The sum of 8269 and 8539 is [[16808edo\|16808]], which does well in not just the 23-limit, but also the 31-limit. The difference between 8269 and 8539 is [[270edo\|270]], which does well in the 13-limit, and also beyond. A step of 8269edo has also been similarly proposed as an [[interval size measure]], the '''major tina'''.

	Some of the simpler commas it [[tempering out\|tempers out]] include [[123201/123200]] in the 13-limit; [[194481/194480]], [[336141/336140]] in the 17-limit; 23409/23408~~, 27456/27455~~, 28900/28899, 43681/43680, 89376/89375 in the 19-limit; and 21505/21504 among others in the 23-limit.		Some of the simpler commas it [[tempering out\|tempers out]] include [[123201/123200]] in the 13-limit; [[194481/194480]], [[336141/336140]] in the 17-limit; 23409/23408, 28900/28899, 43681/43680, 89376/89375 in the 19-limit; and 21505/21504 among others in the 23-limit.

	=== Prime harmonics ===		=== Prime harmonics ===

Overthink: /* Theory */ Added sum and difference edos

2025-10-04T20:53:36Z

Theory: Added sum and difference edos

← Older revision		Revision as of 20:53, 4 October 2025
Line 3:		Line 3:

	== Theory ==		== Theory ==
	8269edo is both a [[The Riemann zeta function and tuning #Zeta edo lists\|zeta peak and zeta integral edo]], which has to do with the fact that it is a very strong [[19-limit\|19-]] and [[23-limit]] system. It has a lower 19-limit and a lower 23-limit [[Tenney-Euclidean temperament measures #TE simple badness\|relative error]] than any smaller division, a lower 19-limit [[Tenney-Euclidean temperament measures #TE simple badness\|TE logflat badness]] than any smaller division, and a lower 23-limit logflat badness than any excepting [[311edo\|311]], [[581edo\|581]], [[1578edo\|1578]] and [[2460edo\|2460]]. While [[8539edo\|8539]] has received most of the attention in this size range, 8269 is actually a bit better in the 23-limit and nearly as good in the 19-limit. They are rather like twins, including the fact both are primes. A step of 8269edo has also been similarly proposed as an [[interval size measure]], the '''major tina'''.		8269edo is both a [[The Riemann zeta function and tuning #Zeta edo lists\|zeta peak and zeta integral edo]], which has to do with the fact that it is a very strong [[19-limit\|19-]] and [[23-limit]] system. It has a lower 19-limit and a lower 23-limit [[Tenney-Euclidean temperament measures #TE simple badness\|relative error]] than any smaller division, a lower 19-limit [[Tenney-Euclidean temperament measures #TE simple badness\|TE logflat badness]] than any smaller division, and a lower 23-limit logflat badness than any excepting [[311edo\|311]], [[581edo\|581]], [[1578edo\|1578]] and [[2460edo\|2460]]. While [[8539edo\|8539]] has received most of the attention in this size range, 8269 is actually a bit better in the 23-limit and nearly as good in the 19-limit. They are rather like twins, including the fact both are primes. The sum of 8269 and 8539 is [[16808edo\|16808]], which does well in not just the 23-limit, but also the 31-limit. The difference between 8269 and 8539 is [[270edo\|270]], which does well in the 13-limit, and also beyond. A step of 8269edo has also been similarly proposed as an [[interval size measure]], the '''major tina'''.

	Some of the simpler commas it [[tempering out\|tempers out]] include [[123201/123200]] in the 13-limit; [[194481/194480]], [[336141/336140]] in the 17-limit; 23409/23408, 27456/27455, 28900/28899, 43681/43680, 89376/89375 in the 19-limit; and 21505/21504 among others in the 23-limit.		Some of the simpler commas it [[tempering out\|tempers out]] include [[123201/123200]] in the 13-limit; [[194481/194480]], [[336141/336140]] in the 17-limit; 23409/23408, 27456/27455, 28900/28899, 43681/43680, 89376/89375 in the 19-limit; and 21505/21504 among others in the 23-limit.

FloraC: Expand the harmonics table a little

2025-10-02T16:04:01Z

Expand the harmonics table a little

← Older revision		Revision as of 16:04, 2 October 2025
Line 8:		Line 8:

	=== Prime harmonics ===		=== Prime harmonics ===
	{{Harmonics in equal\|8269\|columns=9}}		{{Harmonics in equal\|8269\|columns=11}}
	{{Harmonics in equal\|8269\|columns=9\|start=10\|collapsed=true\|title=Approximation of prime harmonics in 8269edo (continued)}}		{{Harmonics in equal\|8269\|columns=11\|start=12\|collapsed=true\|title=Approximation of prime harmonics in 8269edo (continued)}}

	=== Subsets and supersets ===		=== Subsets and supersets ===

FloraC: /* Theory */ extend the harmonics table to the 61-limit

2025-03-11T14:21:18Z

Theory: extend the harmonics table to the 61-limit

← Older revision		Revision as of 14:21, 11 March 2025
Line 3:		Line 3:

	== Theory ==		== Theory ==
	8269edo is both a [[The Riemann zeta function and tuning #Zeta ~~EDO~~ lists\|zeta peak and zeta integral edo]], which has to do with the fact that it is a very strong [[19-limit\|19-]] and [[23-limit]] system. It has a lower 19-limit and a lower 23-limit [[Tenney-Euclidean temperament measures #TE simple badness\|relative error]] than any smaller division, a lower 19-limit [[Tenney-Euclidean temperament measures #TE simple badness\|TE logflat badness]] than any smaller division, and a lower 23-limit logflat badness than any excepting [[311edo\|311]], [[581edo\|581]], [[1578edo\|1578]] and [[2460edo\|2460]]. While [[8539edo\|8539]] has received most of the attention in this size range, 8269 is actually a bit better in the 23-limit and nearly as good in the 19-limit. They are rather like twins, including the fact both are primes. A step of 8269edo has also been similarly proposed as an [[interval size measure]], the '''major tina'''.		8269edo is both a [[The Riemann zeta function and tuning #Zeta edo lists\|zeta peak and zeta integral edo]], which has to do with the fact that it is a very strong [[19-limit\|19-]] and [[23-limit]] system. It has a lower 19-limit and a lower 23-limit [[Tenney-Euclidean temperament measures #TE simple badness\|relative error]] than any smaller division, a lower 19-limit [[Tenney-Euclidean temperament measures #TE simple badness\|TE logflat badness]] than any smaller division, and a lower 23-limit logflat badness than any excepting [[311edo\|311]], [[581edo\|581]], [[1578edo\|1578]] and [[2460edo\|2460]]. While [[8539edo\|8539]] has received most of the attention in this size range, 8269 is actually a bit better in the 23-limit and nearly as good in the 19-limit. They are rather like twins, including the fact both are primes. A step of 8269edo has also been similarly proposed as an [[interval size measure]], the '''major tina'''.

	Some of the simpler commas it [[tempering out\|tempers out]] include [[123201/123200]] in the 13-limit; [[194481/194480]], [[336141/336140]] in the 17-limit; 23409/23408, 27456/27455, 28900/28899, 43681/43680, 89376/89375 in the 19-limit; and 21505/21504 among others in the 23-limit.		Some of the simpler commas it [[tempering out\|tempers out]] include [[123201/123200]] in the 13-limit; [[194481/194480]], [[336141/336140]] in the 17-limit; 23409/23408, 27456/27455, 28900/28899, 43681/43680, 89376/89375 in the 19-limit; and 21505/21504 among others in the 23-limit.

	=== Prime harmonics ===		=== Prime harmonics ===
	{{Harmonics in equal\|8269\|columns=12}}		{{Harmonics in equal\|8269\|columns=9}}
			{{Harmonics in equal\|8269\|columns=9\|start=10\|collapsed=true\|title=Approximation of prime harmonics in 8269edo (continued)}}

	=== Subsets and supersets ===		=== Subsets and supersets ===

ArrowHead294 at 16:32, 20 February 2025

2025-02-20T16:32:55Z

← Older revision		Revision as of 16:32, 20 February 2025
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	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro}}		{{ED intro}}

	== Theory ==		== Theory ==

ArrowHead294 at 16:20, 20 February 2025

2025-02-20T16:20:49Z

← Older revision		Revision as of 16:20, 20 February 2025
Line 1:		Line 1:
	{{Infobox ET}}		{{Infobox ET}}
	{{EDO intro~~\|8269~~}}		{{EDO intro}}

	== Theory ==		== Theory ==

FloraC: Sectioning

2025-01-12T10:16:43Z

Sectioning

← Older revision		Revision as of 10:16, 12 January 2025
Line 2:		Line 2:
	{{EDO intro\|8269}}		{{EDO intro\|8269}}

			== Theory ==
	8269edo is both a [[The Riemann zeta function and tuning #Zeta EDO lists\|zeta peak and zeta integral edo]], which has to do with the fact that it is a very strong [[19-limit\|19-]] and [[23-limit]] system. It has a lower 19-limit and a lower 23-limit [[Tenney-Euclidean temperament measures #TE simple badness\|relative error]] than any smaller division, a lower 19-limit [[Tenney-Euclidean temperament measures #TE simple badness\|TE logflat badness]] than any smaller division, and a lower 23-limit logflat badness than any excepting [[311edo\|311]], [[581edo\|581]], [[1578edo\|1578]] and [[2460edo\|2460]]. While [[8539edo\|8539]] has received most of the attention in this size range, 8269 is actually a bit better in the 23-limit and nearly as good in the 19-limit. They are rather like twins, including the fact both are primes. A step of 8269edo has also been similarly proposed as an [[interval size measure]], the '''major tina'''.		8269edo is both a [[The Riemann zeta function and tuning #Zeta EDO lists\|zeta peak and zeta integral edo]], which has to do with the fact that it is a very strong [[19-limit\|19-]] and [[23-limit]] system. It has a lower 19-limit and a lower 23-limit [[Tenney-Euclidean temperament measures #TE simple badness\|relative error]] than any smaller division, a lower 19-limit [[Tenney-Euclidean temperament measures #TE simple badness\|TE logflat badness]] than any smaller division, and a lower 23-limit logflat badness than any excepting [[311edo\|311]], [[581edo\|581]], [[1578edo\|1578]] and [[2460edo\|2460]]. While [[8539edo\|8539]] has received most of the attention in this size range, 8269 is actually a bit better in the 23-limit and nearly as good in the 19-limit. They are rather like twins, including the fact both are primes. A step of 8269edo has also been similarly proposed as an [[interval size measure]], the '''major tina'''.

Francium: +music

2024-11-26T12:23:18Z

+music

← Older revision		Revision as of 12:23, 26 November 2024
Line 11:		Line 11:
	=== Subsets and supersets ===		=== Subsets and supersets ===
	8269edo is the 1037th [[prime edo]].		8269edo is the 1037th [[prime edo]].

			== Music ==
			; [[Francium]]
			* "Details To Follow" from ''Naughty Girl Era'' (2024) – [https://open.spotify.com/track/12kSjGHgwE4J6AMA4Pxd5d Spotify] \| [https://francium223.bandcamp.com/track/details-to-follow Bandcamp] \| [https://www.youtube.com/watch?v=a9HqvhEXvBk YouTube] – totziensmic in 8269edo

FloraC: +comma

2024-11-05T17:08:17Z

+comma

← Older revision		Revision as of 17:08, 5 November 2024
Line 4:		Line 4:
	8269edo is both a [[The Riemann zeta function and tuning #Zeta EDO lists\|zeta peak and zeta integral edo]], which has to do with the fact that it is a very strong [[19-limit\|19-]] and [[23-limit]] system. It has a lower 19-limit and a lower 23-limit [[Tenney-Euclidean temperament measures #TE simple badness\|relative error]] than any smaller division, a lower 19-limit [[Tenney-Euclidean temperament measures #TE simple badness\|TE logflat badness]] than any smaller division, and a lower 23-limit logflat badness than any excepting [[311edo\|311]], [[581edo\|581]], [[1578edo\|1578]] and [[2460edo\|2460]]. While [[8539edo\|8539]] has received most of the attention in this size range, 8269 is actually a bit better in the 23-limit and nearly as good in the 19-limit. They are rather like twins, including the fact both are primes. A step of 8269edo has also been similarly proposed as an [[interval size measure]], the '''major tina'''.		8269edo is both a [[The Riemann zeta function and tuning #Zeta EDO lists\|zeta peak and zeta integral edo]], which has to do with the fact that it is a very strong [[19-limit\|19-]] and [[23-limit]] system. It has a lower 19-limit and a lower 23-limit [[Tenney-Euclidean temperament measures #TE simple badness\|relative error]] than any smaller division, a lower 19-limit [[Tenney-Euclidean temperament measures #TE simple badness\|TE logflat badness]] than any smaller division, and a lower 23-limit logflat badness than any excepting [[311edo\|311]], [[581edo\|581]], [[1578edo\|1578]] and [[2460edo\|2460]]. While [[8539edo\|8539]] has received most of the attention in this size range, 8269 is actually a bit better in the 23-limit and nearly as good in the 19-limit. They are rather like twins, including the fact both are primes. A step of 8269edo has also been similarly proposed as an [[interval size measure]], the '''major tina'''.

	Some of the simpler commas it [[tempering out\|tempers out]] include [[123201/123200]] in the 13-limit; [[194481/194480]], [[336141/336140]] in the 17-limit; 23409/23408, 28900/28899, 43681/43680, 89376/89375 in the 19-limit; and 21505/21504 among others in the 23-limit.		Some of the simpler commas it [[tempering out\|tempers out]] include [[123201/123200]] in the 13-limit; [[194481/194480]], [[336141/336140]] in the 17-limit; 23409/23408, 27456/27455, 28900/28899, 43681/43680, 89376/89375 in the 19-limit; and 21505/21504 among others in the 23-limit.

	=== Prime harmonics ===		=== Prime harmonics ===